Is there any way to animate sprites using just the logic bricks in the Blender Game Engine?
Since nobody seems to answer i'll try.

There are built-in ways to animate sprites from a spritesheet
using the texture properties. Unfortunately this feature is broken in some versions.

There is a python script that can do it.

A rather uncommon way that worked well for me is to put each
frame of your animation on a separate plane(square mesh) and trigger a loop
in "edit object-->change mesh"-actuator with numeric property counter.